Anyone heard of them being DDOS'd? This certainly sounds like ongoing DDOS and protections kicking in (assuming that by "verify my account" you mean "having to solve a captcha").
Lemmy.world has been under repeated attack recently though, and the behaviors you've described match what I see when th service is down. You can see current status and the history of frequent incidents at https://lemmy-world.statuspage.io/.
To relate to your statement about what fails and how, I can say I've seen the failure-modes change as they adapt the setup, and it's a more complex stack than other lemmy instances in order to deal with the attacks and large scale. It degrades in complex ways that are hard to fully reason about unless you're pretty deeply familiar with how things are out together.
I suspect you're seeing a combination of "lemmy world is broken sometimes", "Cloudflare gives weird errors sometimes", and "clients cache things or degrade to unauthenticated connections sometimes". But in any case, seeing lemmy.world be flaky is not weird, it's having a heckuva time.
I noticed Jerboa saying something about my account being verified on lemmy.world in between 502s and other network errors. I've never seen that before, but looks like it's not exclusive to you.
I am also in nine other instances. But I have some communities on this instance that I don't know how to move into other instances in case this one goes kaput.
As an instance owner I can say lemmy.world is a bit of a troublemaker as it will constantly halt and resume federation activities in a very bursty way and cause momentary spikes in resource usage. I have now upgraded the instance enough that it's not really a problem, just an annoying observation.
Having said that - feel free to test run my instance! The only problem I am currently aware of - gmail outright refuses to accept email from it, so verification is impossible. Thanks, Google! Have not had any reports about other providers, though!